Mampaey and Xu1 showed how Cartesian grid representations of curved flow channels, using a zigzag approximation for the walls, can result in substantial numerical flow losses. There are two sources for these losses. The first source arises from changes in flow direction at a zigzag in the grid boundary. Each abrupt direction change is accompanied by a small loss in kinetic energy. The second source of flow loss may arise from poor approximations of fluid momentum advection near a zigzag boundary. If the finite-difference algorithm uses velocity data located in solid regions outside the channel, these values generally contribute to a slowing down of the flow, i.e., result in a loss of energy.
Flow Loss Reduction
Since FLOW-3D uses a Cartesian grid, it is reasonable to ask if it too suffers from numerical flow losses. The answer is no, it does not. The Fractional Area-Volume Obstacle Representation (FAVOR™) method used exclusively in FLOW-3D eliminates zigzag direction changes by smoothly blocking out fractional portions of grid cell faces and volumes. FAVOR™ also has a collection of special algorithms for computing interfacial areas, evaluating wall stresses, enhancing numerical stability, and for computing advection along solid boundaries.
Energy Conservation Example
A simple demonstration of energy conservation in FLOW-3D is provided by a variation of the Mampaey and Xu experiment. In the figure, we show the lower half of a circular channel with fluid located in the left half. The fluid is initially at rest, but gravity is directed downwards causing the fluid to flow to the right side of the channel. In the absence of flow losses, the fluid should reach the same height on the right side as it started from on the left side.
FLOW-3D simulations of this problem show a realistic sloshing distortion of the free surface (figure above) and the center of mass of the fluid rises to nearly its initial height on the right side of the channel indicating little flow loss. This result is all the more remarkable considering the coarse gridding.
Mampaey, F. and Xu, Zhi-An, Simulation and Experimental Validation of Mould Filling, Proc. Modeling of Casting, Welding and Advanced Solidification Processes VII, London, September 10-12, p.3 (1995).